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Diophantine representation of the decimal expansions of $$e$$ and $$\pi$$. (English) Zbl 0984.11007
Summary: Let $$\alpha \in \{\operatorname {e},\pi \}$$, $$\alpha =[\alpha]+\sum _{\kappa =1}^\infty \alpha _\kappa(\beta)\cdot \beta ^{-\kappa}$$ (where $$\beta \in \mathbb N\setminus \{1\}$$ and $$\alpha _\kappa(\beta)\in \{0,1,\dots ,\beta -1\}$$) and $$\zeta \in \{0,1,\dots ,\beta -1\}$$. We describe short Diophantine representations for the predicate $$\alpha _\kappa(\beta)=\zeta$$. The proofs use methods that were developed for the solution of Hilbert’s Tenth Problem.
##### MSC:
 11A63 Radix representation; digital problems 11U05 Decidability (number-theoretic aspects)
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##### References:
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